Erdős Matching (Conjecture) Theorem

Abstract

Let $\mathcal{F}$ be a family of $k$-sized subsets of $[n]$ that does not contain $s$ pairwise disjoint subsets. The Erdős Matching Conjecture, a celebrated and long-standing open problem in extremal combinatorics, asserts the maximum cardinality of $\mathcal{F}$ is upper bounded by $\max \left\{\left(\genfrac{}{}{0ex}{}{sk-1}{k}\right),\left(\genfrac{}{}{0ex}{}{n}{k}\right)-\left(\genfrac{}{}{0ex}{}{n-s+1}{k}\right)\right\}$. These two bounds correspond to the sizes of two canonical extremal families: one in which all subsets are contained within a ground set of $sk-1$ elements, and one in which every subset intersects a fixed set of $s-1$ elements. In this paper, we prove the conjecture.

There is an major error in the proof which the author is not able to fix